A function is named continuos if for every two values assigned to the variable in the function, if those two are not too different, then outputs of both them are similar too. In other words, a function is named continuos if for every number X in the domain, the outputs of all values close to X are close to the X output too. It is common to hear that a continuos function is one that can be drawn in a paper without lifting the pencil. Most of usual fuctions are continuos: polynomial functions, trygonometrical functions and their inverse functions, exponential and logarithmic functions, rational functions, are all of the continuos, and there are many others. Continuos functions domains are intervals. Intervals are collections of numbers that for every pair in the collection, every real number between them two is also in the collection. So, functions can only be continuos if the domain are real numbers; otherwhise, pencil would have to be lifted in some points when drawing it. Functions can be continuos on specific points, or in a complete interval, when it is continuos in all the points in the interval. Continuos functions have some interesting characteristics. One of them is that if a function is differentiable in a point, then it must be continuos in that same point. Other characteristic is that there is always a primitive for a continuos function. Primitives are other functions associated to the original one that, when differentiated, results into the given. Something else that might be mentioned is that adding two continuos functions results into a third one that must be continuos too, and so does multiplying or composing two continuos functions.
Continuos Functions.
A function is named continuos if for every two values assigned to the variable in the function, if those two are not too different, then outputs of both them are similar too. In other words, a function is named continuos if for every number X in the domain, the outputs of all values close to X are close to the X output too. It is common to hear that a continuos function is one that can be drawn in a paper without lifting the pencil. Most of usual fuctions are continuos: polynomial functions, trygonometrical functions and their inverse functions, exponential and logarithmic functions, rational functions, are all of the continuos, and there are many others.
Continuos functions domains are intervals. Intervals are collections of numbers that for every pair in the collection, every real number between them two is also in the collection. So, functions can only be continuos if the domain are real numbers; otherwhise, pencil would have to be lifted in some points when drawing it. Functions can be continuos on specific points, or in a complete interval, when it is continuos in all the points in the interval.
Continuos functions have some interesting characteristics. One of them is that if a function is differentiable in a point, then it must be continuos in that same point. Other characteristic is that there is always a primitive for a continuos function. Primitives are other functions associated to the original one that, when differentiated, results into the given. Something else that might be mentioned is that adding two continuos functions results into a third one that must be continuos too, and so does multiplying or composing two continuos functions.